If 2 ^ -1 x + ^ -1 x α then: - Vidyadip

MCQ

If $ \alpha \leq 2\sin^{-1} \text{x} + \cos^{-1} \text{x}\leq \betaα$ then:

A
$ \alpha = 0, \beta = \pi(2)$
B
$ \alpha = 0, \beta = 2\pi$
✓
$ \alpha = 0, \beta = \pi$
D
$ \alpha = 0, \beta = 5\pi$

✓

Answer

Correct option: C.

$ \alpha = 0, \beta = \pi$

We know that, $ -\frac{\pi}{2}\le \sin^{-1}\text{x}\le \frac{\pi}{2}$
Now add $ \frac{\pi}{2}$
each sides $ \frac{\pi}{2}-\frac{\pi}{2}\le \sin^{-1}\text{x}+\frac{\pi}{2}\le \frac{\pi}{2}+\frac{\pi}{2}$
$ \Rightarrow 0\le \sin^{-1}+\dfrac{\pi}{2}\le \pi$
$ \Rightarrow 0\le \sin^{-1}+(\sin^{-1}\text{x}+\cos^{-1}\text{x})\le \pi$ use the identity
$ \sin^{-1}\text{x}+\cos^{-1}\text{x}=\frac{\pi}{2}$
$ \therefore 0\le 2\sin^{-1}\text{x}+\cos^{-1}\text{x}\le \pi$
Hence $ \alpha=0, \beta=\pi$

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